The evolutionarily stable strategy for a given payoff matrix contest, although originally determined in terms of a haploid population, has been shown elsewhere to correspond to an equilibrium of the mean strategy of a diploid population. In this note, the equilibrium is shown to be locally stable for diploid populations. This local stability is demonstrated primarily by relating the behaviour of the perturbed diploid population to one, or in some cases two, associated haploid populations.

In the familiar immigration–birth–death process the events of immigration, birth and death relate to the individual. There are processes in which the whole family and not just an individual migrates. Such population growth models are studied in some detail.

A two-dimensional lattice random walk is studied in which, at each stage, the direction of the next step is correlated to that of the previous step. An exact expression is obtained from the characteristic function for the dispersion matrix of the n-step transition probabilities, and the limiting diffusion equation is derived.

In a planar percolation model, faces of the underlying graph, as well as the sites and bonds, may be viewed as random elements. With this viewpoint, Whitney duality allows construction of a planar dual percolation model for each planar percolation model, which applies to mixed models with sites, bonds, and faces open or closed at random. Using self-duality for percolation models on the square lattice, information is obtained about the percolative region in the mixed model.

An index lying between 0 and 1 is presented to describe the degree of liberation (or separation) of one component of a particulate material. It is shown how the index is related to the covariance function of the material, the distribution of shapes and sizes of the particles and to the interaction between fracture surfaces and the structure of the material. The variation of the index with the extent of crushing is investigated, together with the problem of stereological estimation. The index is evaluated for some particular probability models.

We consider the model where φ1, φ2 are real coefficients, not necessarily equal, and the at,'s are a sequence of i.i.d. random variables with mean 0. Necessary and sufficient conditions on the φ 's are given for stationarity of the process. Least squares estimators of the φ 's are derived and, under mild regularity conditions, are shown to be consistent and asymptotically normal. An hypothesis test is given to differentiate between an AR(1) (the case φ1 = φ2) and this threshold model. The asymptotic behavior of the test statistic is derived. Small-sample behavior of the estimators and the hypothesis test are studied via simulated data.

The Poisson shot-noise process discussed here takes the form f:oo H(t, s)N(ds), where N(· is the counting measure of a Poisson process and the H(·, s) are independent stochastic processes. Necessary and sufficient conditions are obtained for convergence in distribution, as t ∼ OC, to any infinitely divisible distribution. The main interest is in the explosive transient one-sided shot-noise, Y(t) = f:1 H(t, s)N(ds) where Var Y(t)∼ oc, Here conditions for asymptotic normality are discussed in detail. Important examples include the Poisson cluster point process and the integrated stationary shotnoise.

We prove that for a diffusion process the first-passage-time p.d.f. through a continuous-time function with bounded derivative satisfies a Volterra integral equation of the second kind whose kernel and right-hand term are probability currents. For the case of the standard Wiener process this equation is solved in closed form not only for the class of boundaries already introduced by Park and Paranjape [15] but also for all boundaries of the type S(I) = a + bt ‘/p (p ∼ 2, a, b E ∼) for which no explicit analytical results have previously been available.

In this paper we study the cyclic behaviour of non-homogeneous Markov systems, i.e. the behaviour of the system under the assumption of periodic sequences of transition matrices, input probabilities, output probabilities and total numbers in the system. We provide a general theorem for the limiting structure of such a system under the cyclic behaviour. We also study the asymptotic stability of non-homogeneous Markov systems and theorems are given which characterize asymptotic stability. An application of the above results is given for a British firm, BS.

It is shown that, in some particular cases, it is equivalent to characterize a continuous distribution by properties of records and by properties of order statistics. As an application, we give a simple proof that if two successive jth record values and associated to an i.i.d. sequence are such that and are independent, then the sequence has to derive from an exponential distribution (in the continuous case). The equivalence breaks up for discrete distributions, for which we give a proof that the only distributions such that Xk, n and Xk+1, n – Xk, n are independent for some k ≧ 2 (where Xk, n is the kth order statistic of X1, ···, Xn) are degenerate.

The existence of ordinary and exponential moments of a point process with conditional intensity of the form is deduced from a Markov chain representation for t – ρN(t). These results form an application of recent theorems of Tweedie (1983a, b) and are used to obtain laws of large numbers for a range of functionals of the process.

We analyze certain self-organizing filing techniques when accesses are assumed to be dependent on each other. The stream of requests for accessing records in a file is modelled as a Markov chain. A general framework is introduced to obtain the asymptotic search cost of a memory-free self-organizing heuristic. The move-to-front heuristic is studied in detail. A formula for the asymptotic search cost, which generalizes that in the case of independent accesses, is obtained. Numerical examples on the performance of the transposition heuristic are provided, and compared with that of the move-to-front heuristic.

This paper gives, in the form of Laplace–Stieltjes transforms and generating functions, the joint distribution of the sojourn time and the number of customers in the system at departure for customers in the general M/G/1 queue with processor sharing (M/G/1/PS).

Explicit formulas are given for a number of conditional and unconditional moments, including the variance of the sojourn time of an ‘arbitrary' customer.

Operator methods are used in this paper to systematically analyze the behavior of the Jackson network. Here, we consider rarely treated issues such as the transient behavior, and arbitrary subnetworks of the total system. By deriving the equations that govern an arbitrary subnetwork, we can see how the mean and variance for the queue length of one node as well as the covariance for two nodes vary in time.

We can estimate the transient behavior by deriving a stochastic upper bound for the joint distribution of the network in terms of a judicious choice of independent M/M/1 queue-length processes. The bound we derive is one that cannot be derived by a sample-path ordering of the two processes. Moreover, we can stochastically bound from below the process for the total number of customers in the network by an M/M/1 system also. These results allow us to approximate the network by the known transient distribution of the M/M/1 queue. The bounds are tight asymptotically for large-time behavior when every node exceeds heavy-traffic conditions.

The model considered in this note has been referred to by Haight (1958), Kingman (1961) and Flatto and McKean (1977) as two queues in parallel. Customers choose the shorter of the two queues which are otherwise independent. This system is known to be inferior to a single queue feeding the two servers, but how much? Some elementary considerations provide a fresh perspective on this awkward boundary-value problem. A procedure is proposed for the solution in the context of finite waiting-room size and some comparisons are made with the single-queue system and an independent two-queue system.

Customers initially enter a service unit via a waiting room. The customers to be served are stored in a service room which is replenished by the transfer of all those in the waiting room at the points in time where the service room becomes empty. At those epochs of transfer, positive random numbers of ‘overhead customers' are also added to the service room. Algorithmically tractable expressions for the stationary distributions of queue lengths and waiting times at various embedded random epochs are derived. The discussion generalizes an earlier treatment by Takács in several directions.

A multitype branching process, the n-family community mating process, is introduced for the purpose of comparing extinction probabilities with those of bisexual Galton–Watson branching processes. Consideration of known properties of standard multitype branching processes leads to conditions which are both necessary and sufficient for extinction in a bisexual Galton–Watson branching process. An application is then made to the counterexample of the author's earlier paper.

Upper and lower bounds are obtained for the integral giving the probability that n invariant chords to a circle all intersect each other.

The similarity between order statistics and record values motivated us, in this paper, to investigate some relationships between the order statistics and the record values. These relationships are employed to characterize a continuous distribution by some moment properties of the spacings of the record values, and hence obtain characterizations of the exponential distribution. Some of the well-known results follow trivially.

This paper considers a problem of determining the optimal number of units for a parallel redundant system. The optimal number N∗ to minimize the expected cost is given by a unique solution of equations. When the system is replaced before failure at time T, the optimal N∗ and T∗ are discussed, and a computing procedure for obtaining these values is specified.